We consider a general k dimensional discounted innite server queues process (alternatively, an Incurred But Not Reported (IBNR) claim process) where the multi-variate inputs (claims) are given by a k dimensional nite state Markov chain and the arrivals follow a renewal process. After deriving a multidimensional integral equation for the moment generating function jointly to the state of the input at time t given the initial state of the input at time 0, asymptotic results for the rst and second (matrix) moments of the process are provided. In particular, when the interarrival or service times are exponentially distributed, transient expressions for the rst two moments are obtained. Also, the moment generating function for the process with deterministic interarrival times is considered to provide more explicit expressions. Finally, we demonstrate the potential of the present model by showing how it allows us to study a semi-Markov modulated innite queues process where the customers (cla 查看全文>>